PGD in computational mechanics

F. Chinesta, A. Leygue

EADS Chair, Ecole Centrale of Nantes

Reduced Bases Workshop

The past

The dream

Computational mechanics

1 – Models defined in high dimensional spaces;

2 – Separating the physical space;

3 – Parametric models: optimization, inverse methods and simulation in real time;

4 – Miscellaneous.

PGD & …

1 - Multidimensional models

Ψ x,t,q1,L ,qN( )

1

11

, , ,..,, , ,..,

( )( )

N

jj

NN

j

tt

t

d

d

Ψ Ψ x q q

x q qq

q

(3 1 3 )N D

Ψ q1,L ,qN( ) ≈ Qi

1 q1( )×L ×i=1

i=M

∑ QiN qN( )

JNNFM, 2006

1

* * *

( , ) ( ) ( ) ( ) ( )

( , ) ( ) ( ) ( ) ( )

n

j jj

u x y X x Y y x yR

u x y R x S y R x y

S

S

*( ) ( ) 0x y

u u u dx dy

M A M A

Successive enrichment:

R S

R⋅S* M (u(R,S,X j ,Yj )) −A( )x×y

dx dy=0•

x

⏐ →⏐ ⏐ S

S R

R* ⋅S M (u(R,S,X j ,Yj )) −A( )x×y

dx dy=0•

y

⏐ →⏐ ⏐ R

Variants:

- Newton linearization;

- Residual minimization in the case of non-symmetric operators;

- Other more efficient and optimal constructors are in progress.

Transient: INNFM 2007

Ψ t,q1,L ,qN( ) ≈ Ti t( )×Qi

1 q1( )×L ×i=1

i=M

∑ QiN qN( )

Non linear models: AR 2007

M Ψn−1( )Ψn =A

Extension of the radial approximation, Ladeveze 1985

Non linear models: CMS 2010

Newton and advanced fixed point M Ψn(s)( )Ψn(s+1) =A

BCF descriptions: MSMSE 2007

Quantum chemistry & Master Equation: IJMCE 2008, EJCM 2010

Complex flows:

Ψ t,x,q1,L ,qN( ) ≈ Ti t( )×Xi x( )×Qi

1 q1( )×L ×i=1

i=M

∑ QiN qN( )

ACME 2009, MCS 2010

2 - Separating the space …

Ψ x, y,z( ) ≈ Xi x( )×Yi y( )×Zi z( )

i=1

i=M

∑

CMAME 2008

Boundary conditions and separating non hexahedral domains

IJNME 2010

u(x, y, z,t)≈ F j ,1(x, y) ⋅F j ,2(z)

j=1

n

∑ ⋅F j ,3(t)

3D modeling in plate domains

CMAME, In press

3D solution with 2D complexity

1

( , , ) ( , ) ( )n

j jj i i

i

u x y z X x y Z z

Shells

xx

yy

zz

FEM solution:

PGD solution

610 dof

500 times steps, implicit

14s using matlab

∂T∂t

−k∂2T∂x2

=S

T (t, x,k)

2

2

TS

tk

x

T

,( )T t x

3 - Parametric modeling

3D

2D

allows the solution of a single problem instead the many solutions required by

usual strategies in:

Optimization & Inverse identification

( , , )T t x k

( , , )dT

t x kdk

is explicitly available

accounting for variability

( , , )g gT t x kk

Parametric solution computed off-line with all the sources of variability considered as extra-coordinates

On-line processing of such solution in real time and using light computing platforms: e.g. smartphones, …

Parametric modeling: MCS 2010, ACME 2010, JNNFM 2010

Process optimization: Composites Part A, 2011

Inverse identification: JNNFM 2011

Control & DDDAS: CMAME, MCS - Submitted

Accounting for variability: CMAME In press

Using smarthphones: CMAME In press, CMAME Submitted

u(x,t, p

1,L , p

Q) ≈ Xi (x) ⋅T(t)

i=1

n

∑ ⋅Pi1 p1( )L Pi

Q pQ( )

Shape optimization: ACME 2010

4 – Miscellaneous Error estimator: CMAME 2010

Coupling PGD & FEM: IJMCE 2011

Coupled models & multi-scale: IJNME 2010, IJMF 2010

Time multi-scale & parallel time integration: IJNME 2011, In press

u =u(τ )

u =u(τ ,t)

PGD based transient boundary element discretizations: EABE 2011

Convective stabilization:- Separating and then stabilizing- Stabilizing and then separating

Computational Mechanics, Submitted